Activists have vandalised the historic home of the 18th century British explorer Captain James Cook with graffiti to protest against Australia's national day.

The walls of Cook's cottage in Melbourne was spray-painted with slogans describing January 26 as shameful.

January 26 is Australia Day and commemorates British settlement of Sydney in 1788. Opponents call it "Invasion Day" and regard it as a shameful reminder that Australian land was taken from Aborigines by British colonists without a treaty.

The cottage was originally built in 1755 in England by Cook's parents, who discovered the site of the first colony.

Detective Senior Constable Scott Gray said it was the third graffiti attack on the building since Australia Day last year.